Method and device for determining a rotation angle or a displacement

ABSTRACT

A method for determining a rotation angle or a path, with the following steps:  
     measurement of at least two phase values α 1 , α 2  through the scanning of sensors that are associated with the respective phase values and have different periodicities,  
     calculation of a working value k, which can be represented as a whole number, on the basis of the measured phase values α 1  and the periodicities n i  associated with them,  
     calculation of at least two scaled estimates φ s1   /2 π on the basis of the phase values α i , the periodicities n i , the working value k, and integral working factors k i  that set the periodicities n i  in relation to one another in a scaling relation, and  
     weighted summation of the scaled estimates φ s1   /2 π in order to obtain a determined estimate Φ meas   /2π.

[0001] The current invention relates to a method and a device for determining an angle of rotation or a path, according to the preambles to claims 1 and 9.

PRIOR ART

[0002] In various applications, particularly in devices intended to aid in determining the angular position of a rotatable shaft, it is desirable to know the precise angular position of the shaft. This requirement can be fulfilled, for example, with the aid of analog angle sensors, for example potentiometers, which after being turned on, immediately output the current angular position value in the form of a voltage in every position.

[0003] If devices of this kind are used for angular measurement of angular ranges greater than 360°, the problem arises that it is no longer possible to determine which rotation the shaft is currently in. In order to evaluate angular ranges that are greater than 360°, however, incremental measuring transmitters can be used, in which the angular position is determined through forward and backward counting of pulses.

[0004] Such incremental measuring transmitters, however, cannot execute absolute angle measurement because it is only possible to count increments that have passed by a receiver.

[0005] In some technical measuring tasks, the use of incremental measuring transmitters generates phase measurement values; the values that are actually to be measured, such as an angle, a path, or a distance, must be determined from these phase measurement values. In order to increase the range of unambiguity (corresponding to a phase range from 0−2π) it is possible to use at least one other measurement conduit with a different phase slope and to derive a greater unambiguity range from a suitable combination of measurement values.

[0006] Examples of this include distance measurement using RADAR or modulated laser light. In these instances, N measurements are executed at different frequencies f₁, . . . f_(N). At the reception point, the signals reflected from a target at a distance x have the following phase shifts (c=the speed of light): $\alpha_{i} = \frac{2 \cdot \pi \cdot f_{i} \cdot 2 \cdot x}{c}$

[0007] The phase shifts are thus proportional to the value that is to be measured and proportional to the frequency used. However, the actual measurement values of the phases always lie in the range from 0 to 2π, i.e. are always determined, with the sole exception of integral multiples of 2π.

[0008] Another example that should be mentioned is an optical angular sensor. In this case, a scan of N optical line gratings is executed, where N tracks with optical line gratings are placed on a disk or a cylinder. There are n_(i) periods or lines in one rotation. If the phase positions of the tracks are measured in relation to a fixed measurement window with the aid of optoelectronic detectors, this yields the following phase lengths:

α_(i)=(n _(i)) modulo (2π).

[0009] The phases are thus proportional to the rotation angle φ and the periodicities. Here, too, the actual measurement values of the phases always lie in the range from 0 to 2π.

[0010] Finally, multiple wave interferometry should also be mentioned. Here, too, for example paths x are measured through the use of at least two different light wavelengths λ_(i), which yields an increased unambiguity range of $\Lambda = {\frac{\lambda_{2} \cdot \lambda_{1}}{\lambda_{2} - \lambda_{1}}.}$

[0011] Here, too, an appropriate dimensioning yields a phase path of the kind indicated above.

[0012] The evaluation of the signals obtained with methods of this kind, i.e. the determination of x and φ, is carried out using vernier methods.

[0013] In the classic vernier method, the difference is calculated between two phase signals, where for the case in which this difference is less than zero, 2π is added. This method has significant limitations since measurement errors in the phases have full impact on the end result. Furthermore, a method of this kind functions only if the two periodicities under consideration differ by precisely 1.

[0014] DE-OS 195 06 938 has disclosed a modified vernier method in which the value of a variable to be measured is determined from two phase signals through weighted addition and through the further addition of an angle range-dependent constant. This method excels in its capacity for significantly reducing measurement errors in the phase signals. For this method, too, however, it is necessary that the two periodicities under consideration differ by precisely 1.

[0015] Finally, DE-P 1004260 has disclosed a method for determining a rotation angle or distance through the evaluation of phase measurement values. In this method, the phase values that are measured in an N-dimensional space are mapped as N−1 new signals S_(i) by means of a linear transformation A. These signals S_(i) are transformed with the aid of a quantizing mechanism into corresponding integral values W_(i) and are converted by means of a linear mapping C into N real values Z_(i). Weighted phase measurement values λ_(i) modulo 2π are added to these values, yielding N estimates for the angle π to be measured. The N estimates are corrected if need be at their discontinuity points and are added up in a weighted fashion taking their phase angles into account.

[0016] The object of the invention is to supply, through the simplest means possible, measurement values for distances x and angles φ on the basis of at least two phase measurement values. It should as a result no longer be necessary to abide by the requirement in the conventional method that the two periodicities must differ by precisely 1.

[0017] This object is attained by means of a method with the features of claim 1 and a device with the features of claim 9.

ADVANTAGES OF THE INVENTION

[0018] The invention provides for a particularly simple method, which can reliably determine measurement values that are to be determined, e.g. an angle φ or a path or distance x. By contrast with conventional methods, there is a large degree of freedom in the selection of periodicities for the determination of at least two phase signals. The weighting of the individual measurement values or estimates in the manner that is provided according to the invention has turned out to be particularly easy to execute, computationally speaking.

[0019] Advantageous embodiments of the method according to the invention and the device according to the invention are the subject of the dependent claims.

[0020] According to a first embodiment of the method according to the invention, for the case in which two phase values α₁, α₂ are supplied, the working value k, which is used in the process of determining a rotation angle or a path, is calculated by rounding the term: $\left\lbrack \frac{{\alpha_{1} \cdot n_{2}} - {\alpha_{2} \cdot n_{1}}}{2 \cdot \pi} \right\rbrack.$

[0021] In this case, two phase values α₁, α₂ are evaluated, which are respectively obtained from sensors or measuring transmitters that each have n₁ and n₂ periods. Computationally speaking, it is easy to carry out the generation and use of such a working value based on two phase values. In this instance, the rounding is the replacement of the calculated value with the nearest smaller or larger whole number. The deviation of the calculated term from the nearest whole number is a measure of the achievable precision of the method.

[0022] A scaling relation between the periodicities n₁, n₂ is suitably selected to be an equation with the form

k ₂ ·n ₁ −k ₁ ·n ₂=1

[0023] Of the infinite number of solution pairs k₁, k₂, the one with the smallest numerical values is advantageously used.

[0024] According to a particularly preferable embodiment of the method according to the invention, the at least two scaled estimates are calculated in the form ${\frac{\Phi \quad s_{i}}{2 \cdot \pi} = \frac{\frac{\alpha_{i}}{2 \cdot \pi} + {k \cdot k_{i}}}{n_{i}}},$

[0025] where i=1, 2 . . . N, and k=the working value.

[0026] It is also preferable that the weighted summation of the at least two scaled estimates for obtaining a determined estimate Φ_(meas) be carried out in the form $\frac{\Phi_{meas}}{2 \cdot \pi} = {\sum\limits_{i = 1}^{N}\left\lbrack {\frac{\Phi \quad s_{i}}{2 \cdot \pi} \cdot g_{i}} \right\rbrack_{{mod}{(1)}}}$

[0027] where the g_(i) 's (i=1, 2 . . . N) represent the weighting factors for which Σg_(i)=1.

[0028] The calculated sum must be taken as modulo 1 (i.e. only the number of decimals need be taken into consideration). It turns out that the resulting estimate Φ_(meas) for the angle Φ to be measured can be produced in a very precise and reliable fashion.

[0029] In the case of two phase measurement values, it is suitable to set the weighting factors g₁, g₂ each equal to 0.5. This setting turns out to be sufficiently precise for a large number of uses.

[0030] According to another preferred embodiment of the method according to the invention, an improvement in the precision of the estimate Φ_(meas) with similarly probable additive measurement errors in α₁ and α₂ is achieved if the weighting factors are produced in the form: $g_{i} = {\frac{n_{i}^{2}}{\sum\limits_{1}^{N}n_{i}^{2}}.}$

[0031] This weighting is optimal for the purpose of a minimal square of error. This weighting turns out to be very suitable particularly for systems with more than two phase signals to be evaluated, i.e. for N>2.

[0032] According to another preferred embodiment, the weighting factors are produced in the form ${g_{i}^{\prime} = {{\frac{n_{i} \cdot w_{i}}{2^{q}}\quad {with}\quad {\sum\limits_{i = 1}^{N}{w_{i}n_{i}}}} = 2^{q}}},$

[0033] where the w_(i)'s represent whole numbers and are selected in such a way that the weights g′_(i) come as close as possible to the ideal weight g_(i). The natural number q here determines the achievable precision. This selection of the weighting factors turns out to be particularly easy to execute in terms of computation since in the generation of this kind of weighting factors, the division by the term $\sum\limits_{1}^{N}n_{i}^{2}$

[0034] or the multiplication by the reciprocal value of this term is eliminated. Only a division by the power of two 2^(q) is required here, which in a two's complement depiction, can be achieved computationally through a simple digit shift by q places to the right. The summation is suitably calculated with q bits, without regard to subtotal arithmetic overflows.

[0035] Preferred embodiments of the invention will now be explained in detail in conjunction with the accompanying drawings.

[0036]FIG. 1 is a diagram for depicting an estimate Φ_(meas) that can be generated on the basis of two phase signals α₁, α₂ by means of the method according to the invention,

[0037]FIG. 2 is a block circuit diagram of a first preferred embodiment of the device or circuit according to the invention, which can execute the method according to the invention,

[0038]FIG. 3 shows a preferred embodiment of a circuit for producing weighted sums for a system with N phase signals, and

[0039]FIG. 4 shows a flowchart to further explain a preferred embodiment of the method according to the invention.

[0040] First, the method according to the invention will be explained in conjunction with FIG. 1 in an example of two phase measurement values α₁, α₂ (N=2) being considered simultaneously.

[0041] A sensor, for example an optical angle sensor with two tracks, delivers two measured phase values α₁, α₂. The two tracks have an n₁-periodic or n₂-periodic structure. The values n₁ and n₂ are the periodicities of the individual tracks. It should first be assumed that n₁ and n₂ are prime. These structures can be achieved, for example, through the use of intrinsically known line gratings. The phase values α₁ and α₂ in the top two rows in FIG. 1 are plotted by way of example for n₁=13 and n₂=17 in relation to the actual angle Φ to be measured (each scaled to 1 through division by 2π). If the mechanical rotation angle Φ to be determined passes through a rotation (360° or 2π), then the phase values α₁ and α₂ pass n₁ and n₂ times through their respective measuring range.

[0042] Based on the phase values α₁, α₂, which as a rule are in digital form, a whole number k is determined as a working value in a first step according to the equation:

k=the rounding of $\begin{matrix} {k = {{the}\quad {rounding}\quad {{of}\quad\left\lbrack \frac{{\alpha_{1} \cdot n_{2}} - {\alpha_{2} \cdot n_{1}}}{2 \cdot \pi} \right\rbrack}}} & \left( {{equation}\quad (1)} \right) \end{matrix}$

[0043] The term “rounding” is understood here to mean the rounding up or down to the nearest whole number. In a second step, two scaled estimates φ_(s1)/2π and φ_(s2)/2π are determined for the value φ to be determined: $\begin{matrix} {{\frac{\Phi \quad S_{1}}{2 \cdot \pi} = \frac{\frac{\alpha_{1}}{2 \cdot \pi} + {k \cdot k_{1}}}{n_{1}}}{\frac{\Phi \quad S_{2}}{2 \cdot \pi} = {\frac{\frac{\alpha_{2}}{2 \cdot \pi} + {k \cdot k_{2}}}{n_{2}}.}}} & \left( {{equation}\quad (2)} \right) \end{matrix}$

[0044] In this case, k₁ and k₂ are fixed factors, which only have to be determined once for given periodicities. More precisely stated, k₁ and k₂ are arbitrary integral solutions of the equation

k ₂ ·n ₁ −k ₁ ·n ₂=1.  (equation (3))

[0045] Of the infinite number of solutions, for example the ones with the lowest numerical values can be assumed. For example, for the periodicities n₁=13 and n₂=17, logical solutions are k₁=3 and k₂=4.

[0046] In a third step, a weighted summation of the scaled estimates with the weighting factors g₁ and g₂ is executed, where after the summation, the result modulo 1 is to be taken (i.e. only the number of decimals). The final estimate Φ_(meas) for the angle Φ to be determined is generated according to the equation $\begin{matrix} {\frac{\Phi_{meas}}{2 \cdot \pi} = {\left\lbrack {{\frac{\Phi_{s1}}{2 \cdot \pi} \cdot g_{1}} + {\frac{\Phi_{s2}}{2 \cdot \pi} \cdot g_{2}}} \right\rbrack_{{mod}{(1)}}.}} & \left( {{equation}\quad (4)} \right) \end{matrix}$

[0047] For the weighting factors, the equation g_(i)+g₂=1 must apply. In many applications, weighting factors of this kind are each set equal to 0.5. An improvement with similarly probable additive measurement errors in α₁ and α₂ is achieved if one uses the weighting according to the equation: $\begin{matrix} {{g_{i} = \frac{n_{i}^{2}}{\sum\limits_{1}^{N}n_{i}^{2}}},} & \left( {{equation}\quad (5)} \right) \end{matrix}$

[0048] which is optimal for the purpose of a minimal square of error. N is the number of simultaneously considered phase measurement values; in the current example, N=2.

[0049] The first two rows in FIG. 1 have already been explained. The third row depicts an estimate Φ_(meas) determined with the method according to the invention for the above-mentioned example n₁=13 and n₂=17, plotted over the mechanical angle Φ.

[0050] The weighting mentioned above has the advantage that the determination of the scaled estimates φ_(s1)/2π and ^(SM) _(s2)/2π does not explicitly have to be carried out. It is sufficient to determine the respective numerators of the fractions on the right side of equation (2). It is no longer necessary to divide by n₁ and n₂ since the weighting factors g_(i) in equation (5) contain the terms n₁ and n₂ ² in the numerator.

[0051] A particularly simple embodiment is obtained if the weighting factors undergo a further slight variation so that it is also no longer necessary to divide by $\begin{matrix} {\sum\limits_{1}^{N}n_{i}^{2}} & \left( {{equation}\quad (6)} \right) \end{matrix}$

[0052] or to multiply by the reciprocal value. To this end, $\begin{matrix} {g_{i}^{\prime} = {{\frac{n_{i} \cdot w_{i}}{2^{q}}\quad {with}\quad {\sum\limits_{1}^{N}{w_{i} \cdot n_{i}}}} = 2^{q}}} & \left( {{{equations}\quad (7)},(8)} \right) \end{matrix}$

[0053] are selected, with whole numbers w_(i) so that the new weights g′_(i) come as close as possible to the ideal weights g_(i) according to equation (5). The natural number q here determines the achievable precision.

[0054] With a weighting of this kind, all divisions are eliminated; all that is required is a division by the power of two 2^(q), which in a two's complement depiction, is achieved through a simple digit shift by q places to the right. The summation is calculated with q bits, without regard to subtotal arithmetic overflows.

[0055] In the example mentioned above (n₁=13, n₂=17), for q=12 bits, the whole numbers w₁=115 and w₂=153 can be used. The resulting weighting factors thus diverge by approx. 1% from the ideal value according to equation (5). Precisely stated, $\begin{matrix} \begin{matrix} {{g_{1}^{\prime} + g_{2}^{\prime}} = {\frac{w_{1} \cdot n_{1}}{2^{q}} + \frac{w_{2} \cdot n_{2}}{2^{q}}}} \\ {= {\frac{115 \cdot 13}{4096} + \frac{153 \cdot 17}{4096}}} \\ {= \frac{4096}{4096}} \\ {= 1} \end{matrix} & \left( {{equation}\quad (9)} \right) \end{matrix}$

[0056] If n₁ and n₂ are not prime numbers so that they share a common divisor L, where

L=GGT(n ₁ ,n ₂)>1,  (equation (10))

[0057] then the method described above can also be used. In lieu of the periodicities, the values divided by L must then used. As a result in this case, this yields an estimate for L-times the angular value Φ.

[0058]FIG. 2 shows a preferred circuit for executing the method according to the invention. For illustrative purposes, please see FIG. 4 in this connection, which depicts a flowchart of the embodiment of the method according to the invention that is executed with the circuit.

[0059] The sensor labeled 10 in FIG. 2 supplies the two phase values α₁ and α₂ (step 401). In multiplication elements 11 and 12, these measurement signals, which are scaled through division by 2π, are multiplied by the periodicities of the individual tracks of the sensor 10, n₂ and n₁ (step 402). The summation according to equation (1) is executed in a summation element (step 403). The rounding according to equation (1) then takes place in a rounding element 14 (step 404). In order to obtain the two scaled estimates φ_(s1)/2π and φ_(s2)/2π, the working value k generated by means of the rounding element 14 is first multiplied by the values k₁ and k₂. As has been mentioned above, the values k₁ and k₂ can be selected according to equation (3). In order to obtain the respective numerator on the right side of equation (2), the phase values α₁/2π and α₂/2π are added to the outputs of the multiplication elements 15 and 16 in summation elements 17 and 18. The execution of the calculating steps according to equation (2) takes place in step 405 in the flowchart. The multiplication by the weights g_(i) according to equation (5) takes place in multiplication elements 19 and 20 (step 406). The output values of the multiplication elements 19 and 20 are supplied to an adding element 21, by means of which the value Φ_(meas)/2π according to equation (4) can be determined (step 407).

[0060]FIG. 3 shows a circuit for producing the weighted sum with the weights g′_(i) according to equation (7) for a system with N phase signals. It is evident that the respective scaled estimates φ_(s1)/2π are first multiplied by the respective periodicities n_(i). The respective multiplication by the whole numbers w_(i) is carried out in subsequent multiplication elements 30. The output values of the multiplication elements 30 are added up in a summation element 31; the resulting sum is multiplied (according to equation (7)) by the value 2^(−q) in a multiplication element 32. The value thus obtained is the value Φ_(meas)/2π according to equation (4) with the weights g according to equation (7).

[0061] The method according to the invention can achieve the advantages of the modified vernier method mentioned in the introduction to the specification and these advantages are further improved by the weighting proposed according to the invention. 

1. A method for determining a rotation angle or a path, with the following steps: generation of at least two phase values α₁ (i=1, 2 . . . N) through the scanning of sensors (10) that are associated with the respective phase values and have different periodicities n_(i), calculation of a working value k, which can be represented as a whole number, on the basis of the measured phase values α₁ and the periodicities n_(i) associated with them, calculation of at least two—in particular scaled—estimates φ_(s1)/2π on the basis of the phase values α_(i), the periodicities n_(i), the working value k, and integral working factors k_(i) that set the periodicities n_(i) in relation to one another in a scaling relation, and weighted summation of the estimates φ_(s1)/2π in order to obtain a determined estimate Φ_(meas)/2π, which represents the rotation angle or path to be determined.
 2. The method according to claim 1, characterized in that for the case in which two phase values α₁, α₂ are supplied, the working value k is calculated by rounding the term: $\left\lbrack \frac{{\alpha_{1} \cdot n_{2}} - {\alpha_{2} \cdot n_{1}}}{2 \cdot \pi} \right\rbrack$

where n₁ is the periodicity of the sensor (10) associated with the phase value α₁, and n₂ is the periodicity of the sensor (10) associated with the phase value α₂.
 3. The method according to one of claims 1 or 2, characterized in that for the case in which two phase values α₁, α₂ are supplied, the scaling relation of the periodicities n₁, n₂ is selected as k₂·n₁−k₁·n₂=1.
 4. The method according to one of the preceding claims, characterized in that the scaled estimates are calculated according to an equation with the form: $\frac{\Phi \quad s_{i}}{2 \cdot \pi} = {\frac{\frac{\alpha_{i}}{2 \cdot \pi} + {k \cdot k_{i}}}{n_{i}}.}$


5. The method according to one of the preceding claims, characterized in that the weighted summation of the at least two scaled estimates is calculated according to an equation with the form: $\frac{\Phi_{meas}}{2 \cdot \pi} = {\sum\limits_{i = 1}^{N}\left\lbrack {\frac{\Phi \quad s_{i}}{2 \cdot \pi} \cdot g_{i}} \right\rbrack_{{mod}{(1)}}}$

where the g_(i)'s represent weighting factors for which the following equation applies: ${\sum\limits_{i = 1}^{N}g_{i}} = 1.$


6. The method according to claim 5, characterized in that for the case in which two phase values α₁, α₂ are supplied, the weighting factors g₁=g₂=0.5 are used.
 7. The method according to claim 5, characterized in that the weighting factors are calculated according to a relation with the form: $g_{i} = {\frac{n_{i}^{2}}{\sum\limits_{1}^{N}n_{i}^{2}}.}$


8. The method according to claim 5, characterized in that the weighting factors g_(i) are calculated according to a relation with the form: ${g_{i}^{\prime} = {{\frac{n_{i} \cdot w_{i}}{2^{q}}\quad {with}\quad {\sum\limits_{i = 1}^{N}\quad {w_{i} \cdot n_{i}}}} = 2^{q}}},$

where the w_(i)'s represent whole numbers, which are selected in such a way that the g_(i)′'s approximate the g_(i) as closely as possible.
 9. A device for determining a rotation angle or a path, with means for generating at least two phase values α₁ (i=1, 2, . . . N) through the scanning of sensors (10) that are associated with the respective phase values and have different periodicities n_(i), means (13, 14) for calculating a working value k, which can be represented as a whole number, on the basis of the measured phase values α₁ and the periodicities n_(i) associated with them, means (15, 16, 17, 18) for calculating at least two—in particular scaled—estimates φ_(s1)/2π on the basis of the phase values α_(i), the periodicity n_(i), the working value k, and integral working factors k_(i) that set the periodicities n_(i) in relation to one another in a scaling relation, and means (21) for the weighted summation of the scaled estimates in order to obtain a determined estimate φ_(meas)/2π, which represents the rotation angle or path to be determined.
 10. A method for determining a rotation angle or a path, with the following steps: generation of at least two phase values α₁ (i=1, 2 . . . N) through the scanning of sensors (10) that are associated with the respective phase values and have different periodicities n_(i), calculation of a working value k, which can be represented as a whole number, on the basis of the measured phase values α₁ and the periodicities n_(i) associated with them, calculation of at least two—in particular scaled—estimates φ_(s1)/2π on the basis of the phase values α_(i), the periodicities n_(i), the working value k, and integral working factors k_(i) that set the periodicities n_(i) in relation to one another in a scaling relation, and weighted summation of the estimates φ_(s1)/2π in order to obtain a determined estimate Φ_(meas)/2π, which represents the rotation angle or path to be determined.
 11. The method according to claim 10, characterized in that for the case in which two phase values α₁, α₂ are supplied, the working value k is calculated by rounding the term: $\left\lbrack \frac{{\alpha_{1} \cdot n_{2}} - {\alpha_{2} \cdot n_{1}}}{2 \cdot \pi} \right\rbrack$

where n₁ is the periodicity of the sensor (10) associated with the phase value α₁, and n₂ is the periodicity of the sensor (10) associated with the phase value α₂.
 12. The method according to claim 10, characterized in that for the case in which two phase values α₁, α₂ are supplied, the scaling relation of the periodicities n₁, n₂ is selected as k₂·n₁−k₁·n₂=1.
 13. The method according to claim 10, characterized in that the scaled estimates are calculated according to an equation with the form: $\frac{\Phi \quad s_{i}}{2 \cdot \pi} = {\frac{\frac{\alpha_{i}}{2 \cdot \pi} + {k \cdot k_{i}}}{n_{i}}.}$


14. The method according to claim 10, characterized in that the weighted summation of the at least two scaled estimates is calculated according to an equation with the form: $\frac{\Phi_{meas}}{2 \cdot \pi} = {\sum\limits_{i = 1}^{N}\quad \left\lbrack {\frac{\Phi \quad s_{i}}{2 \cdot \pi} \cdot g_{i}} \right\rbrack_{{mod}{(1)}}}$

where the g_(i)'s represent weighting factors for which the following equation applies: ${\sum\limits_{i = 1}^{N}g_{i}} = 1.$


15. The method according to claim 14, characterized in that for the case in which two phase values α₁, α₂ are supplied, the weighting factors g₁=g₂=0.5 are used.
 16. The method according to claim 14, characterized in that the weighting factors are calculated according to a relation with the form: $g_{i} = {\frac{n_{i}^{2}}{\sum\limits_{1}^{N}\quad n_{i}^{2}}.}$


17. The method according to claim 14, characterized in that the weighting factors g_(i) are calculated according to a relation with the form: ${g_{i}^{\prime} = {{\frac{n_{i} \cdot w_{i}}{2^{q}}\quad {with}\quad {\sum\limits_{i = 1}^{N}\quad {w_{i} \cdot n_{i}}}} = 2^{q}}},$

where the w_(i)'s represent whole numbers, which are selected in such a way that the g_(i)′'s approximate the g_(i) as closely as possible.
 18. A device for determining a rotation angle or a path, with means for generating at least two phase values α₁ (i=1, 2, . . . N) through the scanning of sensors (10) that are associated with the respective phase values and have different periodicities n_(i), means (13, 14) for calculating a working value k, which can be represented as a whole number, on the basis of the measured phase values α₁ and the periodicities n_(i) associated with them, means (15, 16, 17, 18) for calculating at least two—in particular scaled—estimates φ_(s1)/2π on the basis of the phase values α_(i), the periodicity n_(i), the working value k, and integral working factors k_(i) that set the periodicities n_(i) in relation to one another in a scaling relation, and means (21) for the weighted summation of the scaled estimates in order to obtain a determined estimate Φ_(meas)/2π, which represents the rotation angle or path to be determined. 